Question 1
Yes. As long as the ability on the enhancement can target the card type causing the removal of your last character (even if the removing card is already out of play), you may play it.
Question 2
They are only ever evil. You may not target as neutral. See the FAQ below:
10. How can Covenants and Curses be targeted?
Covenants and Curses are both enhancements and artifacts (Covenants are also good, and Curses are also evil) at face value (while not face-up on the table or face-down in an artifact pile). So, while they are in deck, hand, or discard pile, they can be targeted as enhancements or artifacts. Once they are played, however, they lose their other identity. Therefore, a Covenant or Curse that is played or held in play as an enhancement can no longer be targeted as an artifact; a Covenant or Curse that is played, held, activated, or placed in the artifact pile as an artifact can no longer be targeted as an enhancement. Covenants are good covenants and Curses are evil curses for the purpose of targeting no matter where they are located or how they were played. If a Covenant or Curse is returned to face value (is back in deck, hand, or discard pile), it is once again both an enhancement and artifact unless played again.
Question 3
Yes. As long as the ability on the enhancement can target the card type causing the removal of your last character (even if the removing card is already out of play), you may play it.
Question 4
In a sense. As long as the ability on the enhancement can target the card type causing the removal of your last character (even if the removing card is already out of play), you may play it.